Question

Starting from rest, a uniform solid cylinder rolls without slipping down a ramp inclined at angle theta to the horizontal. Find an expression for its speed after it has gone a distance d along the incline. Your expression should be in terms of the given variables and any other known constants such as acceleration due to gravity, g.

Answer 1

let d is the distance travelled along the incline and theta is the angle of inclination.

initial vertical height of the cyllinder, h = d*sin(theta)

let R is the radius of the cyllinder, w is the angular speed and v is the linear speed of the cyllinder at the bottom of the cyllinder.

now use conservaton of energy

final kinetic energy = initial potential energy

(1/2)*m*v^2 + (1/2)*I*w^2 = m*g*h

(1/2)*m*v^2 + (1/2)*(1/2)*m*R^2*w^2 = m*g*h

(1/2)*m*v^2 + (1/4)*m*R^2*w^2 = m*g*h

(1/2)*m*v^2 + (1/4)*m*v^2 = m*g*h

(3/4)*m*v^2 = m*g*h

v^2 = 4*g*h/3

v = sqrt(4*g*h/3)

= sqrt(4*g*d*sin(theta)/3) <<<<<<<<<---------------Answer